Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ x y=8 $$

Step-by-Step Solution

Answer

The polar equation is \(r = \sqrt{\frac{16}{\sin 2\theta}}\).

Step 1: Identify the Polar Relationships

In polar coordinates, we use the relationships \(x = r \cos \theta\) and \(y = r \sin \theta\). These are the key transformations needed to convert from Cartesian (x, y) to polar coordinates (r, θ).

Step 2: Substitute Polar Relationships

Substitute \(x = r \cos \theta\) and \(y = r \sin \theta\) into the given equation \(xy = 8\). This gives us \((r \cos \theta)(r \sin \theta) = 8\).

Step 3: Simplify the Equation

Simplify the equation by multiplying the terms on the left side: \(r^2 \cos \theta \sin \theta = 8\).

Step 4: Use Trigonometric Identities

Apply the double angle identity: \(\sin 2\theta = 2 \sin \theta \cos \theta\). Therefore, the equation becomes \(r^2 \frac{1}{2} \sin 2\theta = 8\).

Step 5: Isolate \(r\)

Multiply through by 2 to clear the fraction: \(r^2 \sin 2\theta = 16\). To find \(r\), rewrite it as \(r^2 = \frac{16}{\sin 2\theta}\).

Step 6: Express \(r\) in Terms of Known Quantities

Take the square root of both sides to solve for \(r\): \(r = \sqrt{\frac{16}{\sin 2\theta}}\). This is the polar equation that represents the same relation as \(xy = 8\).

Key Concepts

Cartesian Coordinates

Cartesian coordinates, also known as rectangular coordinates, are a system that helps us locate points using two numbers: the

• x-coordinate
• y-coordinate

These two numbers represent the horizontal and vertical positions of a point on a plane. Imagine a grid where each point is defined by how far along and how far up it is from the origin, which is (0,0). The equation provided, \[ xy = 8 \]is a Cartesian equation. It describes the relationship between x and y on this grid. To solve this problem, we'll convert this Cartesian equation into a polar equation.

Trigonometric Identities

Trigonometric identities are mathematical expressions that relate trigonometric functions to one another. They are vital for simplifying and solving equations involving these functions. In our exercise, we primarily use the identity:

• \( \sin 2\theta = 2 \sin \theta \cos \theta \)

This identity helps us manipulate expressions involving sine and cosine into a simpler form. In particular, it allows us to express products of sine and cosine in terms of a double angle, which facilitates the transformation from Cartesian to polar coordinates.

Double Angle Identity

The double angle identity is a trigonometric identity that can simplify expressions involving angles that are twice as large. It is expressed as:

• \( \sin 2\theta = 2 \sin \theta \cos \theta \)

This identity is helpful for problems where expressions like \cos \theta \sin \theta appear, which happens when we convert Cartesian equations to polar ones. In our example, after substituting the polar coordinates \( \xi = r \cos \theta \) and \( y = r \sin \theta \) into the equation \( xy = 8 \), the expression \( r^2 \cos \theta \sin \theta \) emerges. Using the double angle identity simplifies this to \( \frac{1}{2} r^2 \sin 2\theta \), which makes it easier to solve for \( r \).

Polar Equation Conversion

Converting between coordinate systems can make some mathematical problems easier to solve or understand. In polar coordinates, each point is expressed by its distance

• \( r \)
• angle \( \theta \)

as measured from the origin and the positive x-axis, respectively. To convert the expression \( xy = 8 \) into a polar equation, we start by recognizing the fundamental relationships:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

After substitution and some simplification through our trigonometric identities, we discover that the polar expression is \( r = \sqrt{\frac{16}{\sin 2\theta}} \). This form describes the same graph as the original Cartesian equation but provides a different perspective by highlighting radial distance and angular direction.